Test Rmd math server-side, borrowed from https://rpruim.github.io/s341/S19/from-class/MathinRmd.html

math.Rmd

---
title: "Rending math to HTML in R using katex"
output: 
  html_document:
    katex: true
    self_contained: false
vignette: >
  %\VignetteIndexEntry{Rending math to HTML in R using katex}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---


```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```

<style type="text/css">
td
{
    padding:5px 10px 5px 10px;
}
</style>

## Math inside RMarkdown

In side a text chunk, you can use mathematical notation if you surround it by 
dollar signs `$` for "inline mathematics"  and `$$` for "displayed equations".
**Do not leave a space between the `$` and your mathematical notation**.

Example: `$\sum_{n=1}^{10} n^2$` is rendered as $\sum_{n=1}^{10} n^2$.

Example: `$$\sum_{n=1}^{10} n^2$$` is rendered as $$\sum_{n=1}^{10} n^2$$.

The mathematical typesetting is based on LaTeX, so if you need to search 
for the way to make a particular symbol, include `latex` in your search.
But note: Not all LaTeX macros are available without using additional
packages, and those packages likely will only work if you are creating a PDF.
On the plus side, if you are working in PDF, you can use additional packages
that give much better control and/or easier syntax.


In LaTeX, 

 * macros begin with a backslash (`\`)
 * curly braces (`{` and `}`) are used to surround items that are 
 to be considered as one object from LaTeX's perspective.  
 Without them, usually the next letter or digit will be used, 
 but that isn't usually what you want.  For example 
 `$$\sum_x=1^10 x^2$$` produces 
 $$\sum_x=1^10 x^2$$

### Mathematical Notation

Here are some common mathematical things you might use in statistics


<table>
<tr><td>$x = y$</td><td> `$x = y $` </td> </tr>
<tr><td>$x < y$</td><td> `$x < y $` </td> </tr>
<tr><td>$x > y$</td><td> `$x > y $` </td> </tr>
<tr><td>$x \le y$</td><td> `$x \le y $` </td> </tr>
<tr><td>$x \ge y$</td><td> `$x \ge y $` </td> </tr>
<tr><td>$x^{n}$</td><td> `$x^{n}$` </td> </tr>
<tr><td>$x_{n}$</td><td> `$x_{n}$` </td> </tr>
<tr><td>$\overline{x}$</td><td> `$\overline{x}$` </td> </tr>
<tr><td>$\hat{x}$</td><td> `$\hat{x}$` </td> </tr>
<tr><td>$\tilde{x}$</td><td> `$\tilde{x}$` </td> </tr>
<tr><td>$\frac{a}{b}$ </td><td> `$\frac{a}{b}$`</td> </tr>
<tr><td>$\frac{\partial f}{\partial x}$ </td><td> `$\frac{\partial f}{\partial x}$`</td> </tr>
<tr><td>$\displaystyle \frac{\partial f}{\partial x}$ </td><td> `$\displaystyle \frac{\partial f}{\partial x}$`</td> </tr>
<tr><td>$\binom{n}{k}$ </td><td> `$\binom{n}{k}$`</td> </tr>
<tr><td>$x_{1} + x_{2} + \cdots + x_{n}$</td><td> `$x_{1} + x_{2} + \cdots + x_{n}$` </td> </tr>
<tr><td>$x_{1}, x_{2}, \dots, x_{n}$</td><td> `$x_{1}, x_{2}, \dots, x_{n}$` </td> </tr>
<tr><td>$\mathbf{x} = \langle x_{1}, x_{2}, \dots, x_{n}\rangle$</td><td> `\mathbf{x} = \langle x_{1}, x_{2}, \dots, x_{n}\rangle$` (`\bm` from the `bm` pacakge would be better) </td> </tr>
<tr><td>$x \in A$</td><td> `$x \in A$` </td> </tr>
<tr><td>$|A|$</td><td> `$|A|$` </td> </tr>
<tr><td>$x \in A$</td><td> `$x \in A$` </td> </tr>
<tr><td>$A \subset B$</td><td> `$x \subset B$` </td> </tr>
<tr><td>$A \subseteq B$</td><td> `$x \subseteq B$` </td> </tr>
<tr><td>$A \cup B$</td><td> `$A \cup B$` </td> </tr>
<tr><td>$A \cap B$</td><td> `$A \cap B$` </td> </tr>
<tr><td>$X \sim {\sf Binom}(n, \pi)$</td><td> `$X \sim {\sf Binom}(n, \pi)$` (`sf` for "slide font")</td> </tr>
<tr><td>$\mathrm{P}(X \le x) = {\tt pbinom}(x, n, \pi)$</td><td> `$\mathrm{P}(X \le x) = {\tt pbinom}(x, n, \pi)$` (`tt` for "typewriter type")</td> </tr>
<tr><td>$P(A \mid B)$</td><td> `$P(A \mid B)$` </td> </tr>
<tr><td>$\mathrm{P}(A \mid B)$</td><td> `$\mathrm{P}(A \mid B)$` (`mathrm` for "math roman font"</td> </tr>
<tr><td>$\{1, 2, 3\}$</td><td> `$\{1, 2, 3\}$` </td> </tr>
<tr><td>$\sin(x)$ </td><td> `$\sin(x)$`</td> </tr>
<tr><td>$\log(x)$ </td><td> `$\log(x)$`</td> </tr>
<tr><td>$\int_{a}^{b}$</td><td> `$\int_{a}^{b}$` </td> </tr>
<tr><td>$\left(\int_{a}^{b} f(x) \; dx\right)$</td><td> `$\left(\int_{a}^{b} f(x) \; dx\right)$` </td> </tr>
<tr><td>$\left[\int_{-\infty}^{\infty} f(x) \; dx\right]$</td><td> `$\left[\int_{\-infty}^{\infty} f(x) \; dx\right]$` </td> </tr>
<tr><td>$\left. F(x) \right|_{a}^{b}$</td><td> `$\left. F(x) \right|_{a}^{b}$` </td> </tr>
<tr><td>$\sum_{x = a}^{b} f(x)$ </td><td> `$\sum_{x = a}^{b} f(x)$`</td> </tr>
<tr><td>$\prod_{x = a}^{b} f(x)$ </td><td> `$\prod_{x = a}^{b} f(x)$`</td> </tr>
<tr><td>$\lim_{x \to \infty} f(x)$ </td><td> `$\lim_{x \to \infty} f(x)$`</td> </tr>
<tr><td>$\displaystyle \lim_{x \to \infty} f(x)$ </td><td> `$\displaystyle \lim_{x \to \infty} f(x)$`</td> </tr>
</table>

### Greek Letters

<table>
<tr><td>$\alpha A$</td><td>`$\alpha A$`</td><td>$\nu N$</td><td>`$\nu N$`</td></tr>
<tr><td>$\beta B$</td><td>`$\beta B$`</td><td>$\xi\Xi$</td><td>`$\xi\Xi$`</td></tr>
<tr><td>$\gamma \Gamma$</td><td>`$\gamma \Gamma$`</td><td>$o O$</td><td>`$o O$` (omicron)</td></tr>
<tr><td>$\delta \Delta$</td><td>`$\delta \Delta$`</td><td>$\pi \Pi$</td><td>`$\pi \Pi$`</td></tr>
<tr><td>$\epsilon \varepsilon E$</td><td>`$\epsilon \varepsilon E$`</td><td>$\rho\varrho P$</td><td>`$\rho\varrho P$`</td></tr>
<tr><td>$\zeta Z$</td><td>`$\zeta Z	\sigma \,\!$`</td><td>$\Sigma$</td><td>`$\sigma \Sigma$`</td></tr>
<tr><td>$\eta H$</td><td>`$\eta H$`</td><td>$\tau T$</td><td>`$\tau T$`</td></tr>
<tr><td>$\theta \vartheta \Theta$</td><td>`$\theta \vartheta \Theta$`</td> <td>$\upsilon \Upsilon$</td><td>`$\upsilon \Upsilon$`</td></tr>
<tr><td>$\iota I$</td><td>`$\iota I$`</td><td>$\phi \varphi \Phi$</td><td>`$\phi \varphi \Phi$`</td></tr>
<tr><td>$\kappa K$</td><td>`$\kappa K$`</td><td>$\chi X$</td><td>`$\chi X$`</td></tr>
<tr><td>$\lambda \Lambda$</td><td>`$\lambda \Lambda$`</td><td>$\psi \Psi$</td><td>`$\psi \Psi$`</td></tr>
<tr><td>$\mu M$</td><td>`$\mu M$`</td><td>$\omega \Omega$</td><td>`$\omega \Omega$`</td></tr>
</table>

### Aligning equations

If you want a sequence of aligned equations (often very useful for demonstrating
algebraic manipulation or for plugging values into equations), use `\begin{align*} ... \end{align*}`.  Separate lines with `\\` and use `&` to mark where things should line up.
Note: No dollar signs are needed when you use this method.

#### Example

This
```
\begin{align*}
a & = b \\
X &\sim {\sf Norm}(10, 3) \\
5 & \le 10
\end{align*}
```

produces

\begin{align*}
a & = b \\
X &\sim {\sf Norm}(10, 3) \\
5 & \le 10
\end{align*}